By a result of Huybrechts ([1]}), \(\operatorname{\sf Teich}\)  has only finitely
many connected components. We shall fix the component \({\operatorname{\sf Teich}}^0\) 
containing the parameter point for our initial complex structure, and denote by
\(\widetilde{\Gamma }\) 
the subgroup of finite index in the mapping class group fixing this component.
